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Theorem ecoviass 6246
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
ecoviass.1 𝐷 = ((𝑆 × 𝑆) / )
ecoviass.2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
ecoviass.3 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
ecoviass.4 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
ecoviass.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
ecoviass.6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
ecoviass.7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
ecoviass.8 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
ecoviass.9 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)
Assertion
Ref Expression
ecoviass ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝐷,𝑤,𝑣,𝑢
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑄(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑀(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑁(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecoviass
StepHypRef Expression
1 ecoviass.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5546 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
32oveq1d 5554 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ))
4 oveq1 5546 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
53, 4eqeq12d 2070 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → ((([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ))))
6 oveq2 5547 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
76oveq1d 5554 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ))
8 oveq1 5546 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
98oveq2d 5555 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )))
107, 9eqeq12d 2070 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ))))
11 oveq2 5547 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + 𝐶))
12 oveq2 5547 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 5555 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + 𝐶)))
1411, 13eqeq12d 2070 . 2 ([⟨𝑣, 𝑢⟩] = 𝐶 → (((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))))
15 ecoviass.8 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
16 ecoviass.9 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)
17 opeq12 3578 . . . . 5 ((𝐽 = 𝐿𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
1817eceq1d 6172 . . . 4 ((𝐽 = 𝐿𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
1915, 16, 18syl2anc 397 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
20 ecoviass.2 . . . . . . 7 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
2120oveq1d 5554 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
2221adantr 265 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
23 ecoviass.6 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
24 ecoviass.4 . . . . . 6 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2523, 24sylan 271 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2622, 25eqtrd 2088 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
27263impa 1110 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
28 ecoviass.3 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
2928oveq2d 5555 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
3029adantl 266 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
31 ecoviass.7 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
32 ecoviass.5 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3331, 32sylan2 274 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3430, 33eqtrd 2088 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
35343impb 1111 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
3619, 27, 353eqtr4d 2098 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
371, 5, 10, 14, 363ecoptocl 6225 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  cop 3405   × cxp 4370  (class class class)co 5539  [cec 6134   / cqs 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fv 4937  df-ov 5542  df-ec 6138  df-qs 6142
This theorem is referenced by:  addassnqg  6537  mulassnqg  6539  addasssrg  6898  mulasssrg  6900  axaddass  7003  axmulass  7004
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